Objective Bayesian Analysis And Optimal Design For Degradation Model

With the continuous advancement of technology, the reliability of products in engi-neering, machinery and electronic industry becomes increasingly higher and the products of life tests results with few or no failures. Thus, the traditional accelerate life tests which only record time-to-failure face many challenges. Many researches show that degradation data often provide more information than failure time data in assessing and predicting the reliability of product. At present, reliability assessment using degradation modelling has become one of the hot and difficult research problems in reliability statistics. However, most of the existing researches to proceed statistical analysis for different degradation models are based on maximum likelihood and subjective Bayesian methods. The error of maximum likelihood method will be larger when the degradation data is small. Different priors usually lead to different results when one uses subjective Bayesian method. In this dissertation, for different degradation models we use objective Bayesian method to overcome the crucial problem in small sample degradation test research. We also study the optimal accelerated degradation test design. The contents of this dissertation are as follows:(1) We research the Gamma process and Inverse Gaussian process of degradation models with objective Bayesian method respectively. Reference prior and matching priors for unknown parameters in the Gamma process and Inverse Gaussian process of degra-dation models are derived by using objective Bayesian method, and the properties of corresponding posteriors are also discussed. Then a specified Gibbs sampling procedure is given to obtain the Bayesian estimates of the parameters. A numerical example and a real example (GaAs Lasers) are given to show the effectiveness of the method. It deepens and generalizes the existing results of Xu and Tang (2012).(2) We study the objective Bayesian analysis of constant-stress and step-stress accel-erated degradation tests based on Wiener process models. Firstly, we obtain the reference prior and Jeffreys prior through transformation the parameters of constant-stress and step-stress accelerated degradation tests based on Wiener process models; secondly, we prove that the posterior distributions for reference prior and Jeffreys prior are always proper, and we give a Gibbs sampling procedure for the Wiener process degradation models and then obtain the Bayesian estimates of the parameters; at last, the analysis of simulation and a carbon-film resistors example shows the advantages of the method. It was the first time to use noninformative prior in the study of constant-stress and step-stress accelerated degradation test for the Wiener process models.(3) We study the objective Bayesian analysis of Wiener process degradation model with multiple competing failure modes. Under the assumption that the multiple com-peting catastrophic failure are Weibull distributions, two noninformative priors (reference prior and matching prior) for unknown parameters in the competing degradation model are derived by using objective Bayesian method. Based on two different data types in competing degradation models, we discuss the property of the posterior for noninforma-tive priors and give Gibbs sampling procedures and Bayesian estimates of the parameters respectively. A simulation example is conducted to illustrate the competing degradation model. A real example is presented to analyze an electronic device with two kinds of major failure modes—solder/Cu pad interface fracture (a catastrophic failure) and light intensity degradation (a degradation failure).It is the first time that objective Bayesian method is used to analyze the competing degradation model.(4) Optimal constant-stress accelerated degradation test plans are developed under the assumptions that the degradation characteristic follows a Gamma processes. The test stress levels and the proportion of units allocated to each stress level are determined by D-criterion and V-criterion. The general equivalence theorem (GET) is used to verify that the optimized test plans are globally optimum. In addition, compromise test plans are also studied. Finally, an example is provided to illustrate the proposed method and a sensitivity analysis is conducted to investigate the robustness of optimal plans.